Chapter 1 Discrete Probability Distributions - DIM

⊃⊃22 CHAPTER 1. DISCRETE PROBABILITY DISTRIBUTIONSA BA BAAB∼A B A B AA BFigure 1.7: Basic set operations.PropertiesTheorem 1.1 The probabilities assigned to events by a distribution function on asample space Ω satisfy the following properties:1. P (E) ≥ 0 for every E ⊂ Ω.2. P (Ω)=1.3. If E ⊂ F ⊂ Ω, then P (E) ≤ P (F ).4. If A and B are disjoint subsets of Ω, then P (A ∪ B) =P(A)+P(B).5. P (Ã) =1−P(A) for every A ⊂ Ω.Proof. For any event E the probability P (E) is determined from the distributionm byP (E) =ω∈Em(ω),∑for every E ⊂ Ω. Since the function m is nonnegative, it follows that P (E) is alsononnegative. Thus, Property 1 is true.Property 2 is proved by the equationsP (Ω) = ∑ ω∈Ωm(ω) =1.Suppose that E ⊂ F ⊂ Ω. Then every element ω that belongs to E also belongsto F . Therefore,∑m(ω) ,m(ω) ≤ ∑ω∈E ω∈Fsince each term in the left-hand sum is in the right-hand sum, and all the terms inboth sums are non-negative. This implies thatand Property 3 is proved.P (E) ≤ P (F ) ,